Mathematical Methods and Models in the Natural to the Life SciencesView this Special Issue
Dynamics of a Stochastic Cooperative Predator-Prey System with Beddington-DeAngelis Functional Response
Stochastic cooperative predator-prey system with Beddington-DeAngelis functional response is studied. It presents an investigation of dynamic properties of the system. Our results show that there exists a unique positive solution to the system for any positive initial value, and the positive solution is stochastically bounded. Moreover, under some conditions, we analyze global asymptotic stability of the positive solutions. With small environmental noises, the stochastic system is getting more similar to the corresponding deterministic system. Neither of the species in the system will die out. Finally, simulations are carried out to conform to our result.
As we all know, in mathematical biology, predator-prey system, competitive system, and cooperative system are the three rudimentary and important ecological systems. The dynamic relationship between species has long been and will continue to be a dominant theme in ecology due to its universal existence and importance. It is well-known that predator-prey systems are very important and extensive in the nature fields. One significant component of the predator-prey relationship is predator’s functional response, that is, the rate of prey consumption by an average predator. There are many significant functional responses in order to model various different situations. In fact, most of the functional responses are prey-dependent; however, some biologists have argued that in many cases, especially when predators have to search for food and therefore have to share or compete for food, the traditional predator-prey systems with prey-dependent functional response fail to model the interference among predators, the functional response should be predator-dependent. In [1, 2], Beddington and DeAngelis proposed the following predator-prey model with Beddington-DeAngelis functional response: Skalski and Gilliam  compared statistical evidence from 19 predator-prey systems, and then they claimed that three predator-dependent functional responses (Hassell-Varley, Beddington-DeAngelis, and Growley-Martin) can provide better description of predator feeding over a range of predator-prey abundances. And the Beddington-DeAngelis type functional response was even suitable in some cases.
But most of this work is restricted to predator-prey systems, little has been done for cooperative systems [4, 5]. May  suggested the following set of equations: to describe a pair of mutualists.
However, there is often the interaction among multiple species in nature, whose relationship is more complex than those in two species. Therefore, it is more realistic to consider the multiple-species predator-prey systems. In order to continue studying such models, in this paper, we consider a cooperative predator-prey system with Beddington-DeAngelis functional responses at first: where species is the prey of and , , and are cooperative species. All the parameters in system (3) are positive constants.
In fact, population dynamics is inevitably affected by environmental white noise which is an important component in an ecosystem. But the model (3) is deterministic and does not incorporate the effect of environmental noise. May  also pointed out the fact that due to environmental fluctuation, the birth rates, carrying capacity, competition coefficients, and other parameters involved in system exhibit random fluctuation to a greater or a lesser extent. Therefore many scholars rewrote the deterministic models as stochastic ones subjected to stochastic noises, for studying the effect of environmental variability on the population dynamics [7–9].
The parameters in the real ecosystems are often subject to lots of environmental noises, since they relate to climate, geographical distribution, geological features, human disaster, human intervention, and other environmental factors. Therefore, the logistics and energy flow, in which they are determined by groups, are fluctuating. The oscillation in population biomass is directly manifested as birth and death rates of random perturbation. Currently, one of the main ways considered in the literature to model the effect of the environmental fluctuations in population dynamics is to assume that the most sensitive parameter is the intrinsic growth rate. Thus, in this paper we introduce some stochastic perturbation into the intrinsic growth rate. Therefore, the intrinsic growth rate can be written as an average growth rate adding some small random perturbed terms. In general, by the well-known central limit theorem, the small terms follow some normal distributions, so we can use standard Brownian motions to represent the environmental fluctuations.
In this paper, taking into account the effect of randomly fluctuating environment, we introduce stochastic perturbation into growth rates , , and to become , , and in system, where represents the intensity of the noise and is a standard white noise, namely, is a standard Brownian motion defined on a complete probability space . Then the stochastic system takes the following form: Considering system (4), the initial conditions , , and will be referred to.
2. Global Positive Solutions
Lemma 1. For any initial value , where , system (4) has a unique positive local solution for almost surely, where is the explosion time.
Proof. Consider the following equations: on with initial value , , and . The coefficients of (5) satisfy the local Lipschitz condition, thus there is a unique local solution on . Then , , and are the unique positive local solutions with initial value , , and by Itô’s formula.
Theorem 2. For any initial value , there is a unique solution of system (4) on , and the solution will remain in with probability 1.
Proof. According to Lemma 1, we only need to show that . Let be sufficiently large for , , and lying within the interval . For each integer , we define the stopping times Obviously, is increasing as . Set ; hence, almost surely. Now, we only need to show that . If this statement is false, there is a pair of constants and such that . Thus there exists an integer such that Define a function : by The nonnegativity of this function can be seen from and . If , we have then where is a positive number. Substituting this inequality into (9), we see that Integrating both sides of the above inequality from 0 to and then taking the expectations leads to Set , then we get by inequality (7). Obviously, for every , there are at least , , and which equal either or , then is no less than It then follows from (9) that where is the indicator function of , letting , we have that This completes the proof.
3. Stochastic Boundedness
Definition 3. The solution of system (4) is said to be stochastically ultimately bounded, if for any , there is a positive constant , such that for any initial value , the solution of system (4) has the property that
Assumption A. For any initial value , there exists such that
Lemma 4. Assume that Assumption A holds. Let be a positive solution of (4) with any initial value , for all , then where
Proof. Define the function , for and . By Itô’s formula we get
Integrating from 0 to and taking expectations yields
Let , then we have
From (17), we know
which by the standard comparison argument shows that
Define the function , for and . By Itô’s formula we get Integrating from 0 to and taking expectations yields So, Let , then we have From (17), we know that which by the standard comparison argument shows that that is, Similarly, we can show that This completes the proof.
Theorem 5. Assume that Assumption A holds, the solutions of system (4) with initial value are stochastically ultimately bounded.
Proof. If , its norm here is denoted by , then by Lemma 4, , . is dependent on and is defined by . By virtue of Chebyshev inequality, we can easily obtain that the solution of system (4) is stochastically ultimately bounded.
4. Stochastic Permanence
Definition 6 (see ). The solution of system (4) is said to be stochastically permanent, if for any , there exist a pair of positive constants and such that for any initial value , the solution of system (4) has the properties that
Assumption B. One has
Theorem 7. Under Assumption B, for any initial value , the solution of system (4) satisfies that where is an arbitrary positive constant satisfying and is an arbitrary positive constant satisfying
Proof. Define for , then
Define for , by Itô’s formula, we get
Under Assumption B, choosing a positive constant such that it satisfies (38). By Itô’s formula, we get where then choosing a positive constant such that it obeys (39), by Itô’s formula, where where Hence, it implies that there exists a positive constant such that Then we have Integrating both sides of the above inequality from 0 to and then taking the expectations leads to where . So Since that , where , obviously as required.
Theorem 8. Under Assumption B, system (4) is stochastically permanent.
Proof. By Theorem 5, we know that Now, for any , let . Then by Chebyshev’s inequality, we can obtain the conclusion easily.
5. Global Asymptotic Stability
Definition 9. Let be a positive solution of system (4). If we say that is globally asymptotically stable in expectation, it means that any other solution of system (4) has and that we have initial value . That is
Lemma 10 (see ). Suppose that an n-dimensional stochastic process on satisfies the condition for some positive constants , , and . Then there exists a continuous modification of which has the property that for every there is a positive random variable such that In other words, almost every sample path of is locally but uniformly Hölder continuous with exponent .
Lemma 11. Let be a solution of system (4) on with initial value , then almost every sample path of is uniformly continuous on .
Proof. From system (4), we have the following stochastic integral equation Let − , = , notice that On the other hand, by the moment inequality (see ) for stochastic integrals, we have that for and , where . Let , , and , we obtain where . Then, we have that almost every sample path of is locally but uniformly Hölder continuous with exponent for every and therefore almost every sample path of is uniformly continuous on . Similarly, we can show that almost every sample path of and is uniformly continuous on .
Lemma 12 (see ). Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on . Then .
Theorem 13. If then system (4) is globally asymptotically stable.
Proof. Define then is a continuous positive function on . A direct calculation of the right differential of , and then applying Itô’s formula, we have Integrating from 0 to and taking expectations yields So